A discrete-time dynamical system is given as follows: xn+1=f(xn), n=0,1,2, . . . , N, where xn ∈  and x0 is the initial condition, such dynamical system is usually referred to as map, as it is fully determined by its right hand side. The simplest maps are the so-called uni-modal maps like the logistic and tent maps, while their generalization, the so-called multi-modal or k-modal maps may present even more rich dynamical behaviors.
Definition 1: A map f: ℑ→ℑ is called as the k-modal one, if it is continuous on ℑ and it has k critical points denoted by c0, c1, . . . , ck−1 in ℑ. Moreover, there exist intervals ℑi, i=0, . . . k−1, ∪i=1k ℑi−1=ℑ, such that ∀i=0, . . . , k−1, it holds ci ∈ℑi and ƒβ(ci)>ƒβ(x) ∀x ∈ℑi and x≠ci, where β is a parameter.
The case k=1 will be further simply referred as to the so-called uni-modal map.
To be more specific, ℑ:=[0,1] and recall that a critical point c of the continuous piecewise smooth map f(x)ℑ→ℑ is c ∈ ℑ where f is differentiable and f′(c)=0.
A critical point c occurs for f′(c)=0 or f′(c) does not exist. But continuous smooth maps always present f′(c)=0. The above definition does not constraint a function to have only k critical points. However it only considered those that are local maximum on a subinterval.
Definition 2. Random Number Generator (RNG) is a computer or hardware device designed to produce sequences of numbers without an order apparently.
Definition 3. Pseudo-Random Number Generator (PRNG) is an algorithm that produces a sequence of numbers with an excellent approximation to a set of random numbers.
Definition 4. Cryptographically Secure Pseudo-Random Number Generator (CSPRNG) is a pseudorandom number generator (PRNG) with features that make it suitable for use in cryptography.
Some patents that have implemented this type of significant expertise in the development of new technologies, are depicted herein below:
Typically a Pseudo-Random Bit Generator is based on a Linear Feedback Shift Register (LFSR), this generator consists of a group of flip-flops and exclusive OR, which allow some bits in the register to be true; in other words allowable in the register and selected bits are used, for example, scientific instruments used to measure the intensity of the solar radiation reflected by the earth in various directions and spectral bands; digital cameras are conventionally gathered in spectral bands at different angles for generating images with different wave lengths; the critical points of the above definition, allowing to generate an array representing entries image generation, the function of various wavelengths for these purposes.
A main feature of the proposed invention is that describes an algorithm for generating pseudo-random bits sequences, the usefulness of these sequences includes creating encryption keys and generating a chain-key, which can be equated to addresses spectral bands like LFRS operate; ie holistically, the generator is based on a dynamic discrete time system whose implementation is more versatile than that described above.
In the prior art are some patents related to applications that are contained in this application, such as:
The U.S. Pat. No. 8,738,675B2 patent (Random Number Generator using chaos in continuous time) relates to a method of generating random numbers based on continuous-time chaotic oscillators; where compensation frequency is employed to maximize the statistical quality of an output stream where the proposed circuit can be realized in integrated circuits and the use of continuous chaos in generating random numbers with a very high yield; another invention relates to Patent CN101364171B (real random numbers dynamic generator) relating to a generator, which employs data entered to Von Neumann validators to adjust the output bits by means of chaos conversion algorithms through discrete components and block cipher; especially using the Lyapumov index for the transformation of the block, as a way to control the chaos within a set range and the use of so called S-box that influences the transformation of encrypted blocks;
The patent U.S. Pat. No. 6,954,770 (Random Number Generator) comprising an oscillator with an output signal dependent on a random source which is not based on chaos theory, a sampling device for sampling the output signal of the oscillator to obtain a sampled oscillator output, and a fixed frequency clock driven by linear feedback shift register (LFSR) communicatively coupled to the sampling device through a digital gate to receive the output of the sampled oscillator, and to provide a random number in a LFSR output. Furthermore, the random number generator may comprise an optional communicatively coupled to the LFSR, to read the random number mixing function, and to insert the random number in an algorithm in order to obtain a robust random number; however, this technology depends only on the two oscillators and possible responses that keep only presents these two output options.
The above patents are useful in cryptography for generating keys/cipher/data input in symmetric ciphers. A set of patents that are related architectures chaotic computing logic gates based on nonlinear elements, while the present invention provides dynamically configurable structures linear basic logic gates are: U.S. Pat. Nos. 8,091,062, 7,973,566, 7,924,059, 7,863,937, 7,415,683, 7,096,437, 7,453,285, 7,925,814, 7,925,131 and US 2010/0219858 applications.
All of the above patents represent antecedents either input data, operation data or output data, but the type of related inventions contained in this application are those that relate to random number generators, where his output could be reproduced when the initial condition is known. A variety of such systems, eg U.S. Pat. Nos. 8,023,649B2 and 7,925,014B2 which proposed algorithms pseudo-random generators based on cellular automata.
The pseudo-random generator proposed in this patent is based on the use of dynamic systems due to the intrinsic properties of these systems as are the sensitivity to initial conditions, complex and deterministic behavior, ergodicity, etc.
It includes implementations of dynamic continuous time systems, similar to those of the US2014101217A1, EP2758510A1 and U.S. Pat. No. 7,170,997B2 patents, which required the use of numerical methods for their evolution in time when they are implemented by software, however when implemented with electronic circuits or Field Programmable Gate Array (FPGA) is possible to obtain its evolution through hardware.
In the prior art there are pseudo-random generators based on dynamic discrete-time systems such as US2006251250A1 patent proposes a pseudo-random generator based on mapping systems Hénan and Lozi, the generation method consist on repeatedly iterate the system to generate the array of bits 1 or 0 whose possible dimension is 1×N, each iteration is converted to integer and finally using module 2 each iteration becomes a bit.
The S6064738A patent uses a mapping of two-dimensional (Baker Mapping) to build an array of bits with dimensions N×M, which performs a permutation of pixels, to exchange pixel places is important to note that performing this step the histogram of the image does not change, in addition, an extension of the mapping to three dimensions is made and then the value of the pixels is replaced so that in this way the distribution of pixels is changed , as if it were a function f(x, y) where x ε N and y ε M.
In U.S. Pat. No. 6,014,445A Patent Chebyshev map used to generate sequences with real values and has four methods in order to “binarizing” time series;
a) The first method is based on the use of a threshold
b) The second methodology the absolute value is considered for each iteration, and then it takes its binary representation.
c) The third methodology it is a generalization of the second methodology such that a scaling-dependent interval where the system evolves is performed, so if the interval is [0,1] then the methodology 2 and 3 are equal.
d) In the fourth methodology it is considered M thresholds (these thresholds are the keys) so that each real value of the time series is evaluated by these thresholds and from a real value are obtained M bits, finally combine these M bits by operations XOR and obtain 1 bit per iteration.
The US2013129088A1 patent proposes a pseudo-random generator based on the combination of multiple discrete time dynamical systems, specifically using the cyclic shift map, for this defining k identical maps which are iterated independently then combined by XOR, so that for any iteration n it is previously required to iterate the k maps.
Finally, there have been proposals generators based on the following types:
First, using chaotic uni-modal maps, some of the patents disclose bit-stream pseudorandom generator using only a logistic map;
Second, using a bit string generated through a chaotic system, which is potentially unsafe because the output may leak some information about the chaotic system.
Third, by the use of two chaotic maps which are combined to obtain a complex bit-stream.